Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2201 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23×C3 |
| prime radical | C3×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ G | |
226 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,26)(14,27)(15,31)(16,23)(17,24)(18,32)(19,25)(20,28)(21,30)(22,29)(33,34,35);(1,3,6,14)(2,7,10,21)(4,15,17,12)(5,13,19,30)(8,22,24,20)(9,11,25,27)(16,29,18,28)(23,31,32,26);(1,4,6,17)(2,8,10,24)(3,12,14,15)(5,18,19,16)(7,20,21,22)(9,23,25,32)(11,26,27,31)(13,29,30,28)51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5492);